Identifier
Mp00231: Integer compositions bounce pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00108: Permutations cycle type Integer partitions
Images
=>
Cc0005;cc-rep-1Cc0002;cc-rep-3
[1]=>[1,0]=>[1]=>[1] [1,1]=>[1,0,1,0]=>[1,2]=>[1,1] [2]=>[1,1,0,0]=>[2,1]=>[2] [1,1,1]=>[1,0,1,0,1,0]=>[1,2,3]=>[1,1,1] [1,2]=>[1,0,1,1,0,0]=>[1,3,2]=>[2,1] [2,1]=>[1,1,0,0,1,0]=>[2,1,3]=>[2,1] [3]=>[1,1,1,0,0,0]=>[3,1,2]=>[3] [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[1,2,3,4]=>[1,1,1,1] [1,1,2]=>[1,0,1,0,1,1,0,0]=>[1,2,4,3]=>[2,1,1] [1,2,1]=>[1,0,1,1,0,0,1,0]=>[1,3,2,4]=>[2,1,1] [1,3]=>[1,0,1,1,1,0,0,0]=>[1,4,2,3]=>[3,1] [2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,1,3,4]=>[2,1,1] [2,2]=>[1,1,0,0,1,1,0,0]=>[2,1,4,3]=>[2,2] [3,1]=>[1,1,1,0,0,0,1,0]=>[3,1,2,4]=>[3,1] [4]=>[1,1,1,1,0,0,0,0]=>[4,1,2,3]=>[4] [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5]=>[1,1,1,1,1] [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4]=>[2,1,1,1] [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5]=>[2,1,1,1] [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[1,2,5,3,4]=>[3,1,1] [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,5]=>[2,1,1,1] [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4]=>[2,2,1] [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[1,4,2,3,5]=>[3,1,1] [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[1,5,2,3,4]=>[4,1] [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,1,3,4,5]=>[2,1,1,1] [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,1,3,5,4]=>[2,2,1] [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,5]=>[2,2,1] [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,1,5,3,4]=>[3,2] [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[3,1,2,4,5]=>[3,1,1] [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[3,1,2,5,4]=>[3,2] [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[4,1,2,3,5]=>[4,1] [5]=>[1,1,1,1,1,0,0,0,0,0]=>[5,1,2,3,4]=>[5] [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6]=>[1,1,1,1,1,1] [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,6,5]=>[2,1,1,1,1] [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,4,6]=>[2,1,1,1,1] [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,6,4,5]=>[3,1,1,1] [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,5,6]=>[2,1,1,1,1] [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5]=>[2,2,1,1] [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,5,3,4,6]=>[3,1,1,1] [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,6,3,4,5]=>[4,1,1] [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,4,5,6]=>[2,1,1,1,1] [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,4,6,5]=>[2,2,1,1] [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,6]=>[2,2,1,1] [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,6,4,5]=>[3,2,1] [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,4,2,3,5,6]=>[3,1,1,1] [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,4,2,3,6,5]=>[3,2,1] [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,5,2,3,4,6]=>[4,1,1] [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,6,2,3,4,5]=>[5,1] [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6]=>[2,1,1,1,1] [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,6,5]=>[2,2,1,1] [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,3,5,4,6]=>[2,2,1,1] [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,3,6,4,5]=>[3,2,1] [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,5,6]=>[2,2,1,1] [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5]=>[2,2,2] [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,5,3,4,6]=>[3,2,1] [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,6,3,4,5]=>[4,2] [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,2,4,5,6]=>[3,1,1,1] [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,2,4,6,5]=>[3,2,1] [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[3,1,2,5,4,6]=>[3,2,1] [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,1,2,6,4,5]=>[3,3] [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[4,1,2,3,5,6]=>[4,1,1] [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,1,2,3,6,5]=>[4,2] [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[5,1,2,3,4,6]=>[5,1] [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[6,1,2,3,4,5]=>[6] [1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7]=>[1,1,1,1,1,1,1] [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,6]=>[2,1,1,1,1,1] [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,7]=>[2,1,1,1,1,1] [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,7,5,6]=>[3,1,1,1,1] [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,6,7]=>[2,1,1,1,1,1] [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6]=>[2,2,1,1,1] [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,6,4,5,7]=>[3,1,1,1,1] [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,7,4,5,6]=>[4,1,1,1] [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7]=>[2,1,1,1,1,1] [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,4,3,5,7,6]=>[2,2,1,1,1] [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,7]=>[2,2,1,1,1] [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,7,5,6]=>[3,2,1,1] [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,5,3,4,6,7]=>[3,1,1,1,1] [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,5,3,4,7,6]=>[3,2,1,1] [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,6,3,4,5,7]=>[4,1,1,1] [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,7,3,4,5,6]=>[5,1,1] [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7]=>[2,1,1,1,1,1] [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,7,6]=>[2,2,1,1,1] [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,6,5,7]=>[2,2,1,1,1] [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,7,5,6]=>[3,2,1,1] [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,6,7]=>[2,2,1,1,1] [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6]=>[2,2,2,1] [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,6,4,5,7]=>[3,2,1,1] [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,7,4,5,6]=>[4,2,1] [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,4,2,3,5,6,7]=>[3,1,1,1,1] [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,4,2,3,5,7,6]=>[3,2,1,1] [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,4,2,3,6,5,7]=>[3,2,1,1] [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,4,2,3,7,5,6]=>[3,3,1] [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,5,2,3,4,6,7]=>[4,1,1,1] [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,5,2,3,4,7,6]=>[4,2,1] [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,6,2,3,4,5,7]=>[5,1,1] [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,7,2,3,4,5,6]=>[6,1] [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7]=>[2,1,1,1,1,1] [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,7,6]=>[2,2,1,1,1] [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,6,5,7]=>[2,2,1,1,1] [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,7,5,6]=>[3,2,1,1] [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,5,4,6,7]=>[2,2,1,1,1] [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,5,4,7,6]=>[2,2,2,1] [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,6,4,5,7]=>[3,2,1,1] [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,7,4,5,6]=>[4,2,1] [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7]=>[2,2,1,1,1] [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,3,5,7,6]=>[2,2,2,1] [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,7]=>[2,2,2,1] [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,7,5,6]=>[3,2,2] [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,5,3,4,6,7]=>[3,2,1,1] [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,5,3,4,7,6]=>[3,2,2] [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,6,3,4,5,7]=>[4,2,1] [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,7,3,4,5,6]=>[5,2] [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,1,2,4,5,6,7]=>[3,1,1,1,1] [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[3,1,2,4,5,7,6]=>[3,2,1,1] [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[3,1,2,4,6,5,7]=>[3,2,1,1] [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,7,5,6]=>[3,3,1] [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[3,1,2,5,4,6,7]=>[3,2,1,1] [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[3,1,2,5,4,7,6]=>[3,2,2] [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[3,1,2,6,4,5,7]=>[3,3,1] [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,1,2,7,4,5,6]=>[4,3] [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[4,1,2,3,5,6,7]=>[4,1,1,1] [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[4,1,2,3,5,7,6]=>[4,2,1] [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[4,1,2,3,6,5,7]=>[4,2,1] [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[4,1,2,3,7,5,6]=>[4,3] [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[5,1,2,3,4,6,7]=>[5,1,1] [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[5,1,2,3,4,7,6]=>[5,2] [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[6,1,2,3,4,5,7]=>[6,1] [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7,1,2,3,4,5,6]=>[7] [1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8]=>[1,1,1,1,1,1,1,1] [1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,6,5,7,8]=>[2,2,1,1,1,1] [1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,5,7,6,8]=>[2,2,1,1,1,1] [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,4,2,3,7,5,6,8]=>[3,3,1,1] [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,6,8,7]=>[2,2,1,1,1,1] [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2] [3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,5,8,6,7]=>[3,3,1,1] [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[4,1,2,3,8,5,6,7]=>[4,4]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Map
cycle type
Description
The cycle type of a permutation as a partition.